Passive Acoustic Radiators
Justin Yates,
Wittenberg University
Spring 2014
Outline
• Review of Passive Acoustic Radiator (PAR or
PR)
• Model of PAR system as a damped, driven
oscillator
• Solutions to the system and their physical
meanings
• Designing a PAR
• Questions
Review
• We can model the
passive cone of a PAR as
a mass on a spring
x
k
F(t)
β
• The oscillator then, is
both driven and damped
Review
Masses (m)
• PR’s need to be tuned
• Tuning can be most
easily achieved by
changing the mass
 0
Restoring
spring (k)
k
m
Damped Harmonic Oscillator
Modeling the PR as a damped, driven oscillator gives us the 2nd
order linear differential equation for the position of the cone;
x ' '2  x '02 x  f 0 cos(t )
Eq. (1)
This equation comes from the equation of motion for the driving
force of the oscillator
mx' 'bx ' kx  F (t )
Where I have simplified by dividing by m and substituting
equivalent coefficients.
The driving force in a PAR is actually a sinusoidal function of time
so
F (t )
 f 0 cos(t )
m
Solutions
The homogeneous equation corresponding to Eq. (1) has the
characteristic equation
r 2 2  r  02  0
whose solution takes the form
xc  C1e r1t  C2 e r2t
After finding the roots and simplifying the corresponding
solution is
xc  A' e  t cos(1   ' )
Solutions
The homogeneous equation corresponding to Eq. (1) has the
particular solution of the form
x p  Ceit
The complex part can be drawn out into the coefficients, let
C  Ae  i
and multiply C by its conjugate, C* so when solved for A2 gives
f 02
A 
(02   2 ) 2  4 
2
and thus the particular solution is
x p  A cos(1   )
Discussion
The general solution is
x(t )  A cos(t   )  A' e  t cos(1   ' )
Harmonic
Transient
It is important to note that the solution consists of two distinct
parts. Of the two terms, one is transient and the other is
harmonic, or steady.
Physical Meaning
The full solution
x (t )  A cos(t   )  A' e  t cos(1t   ' )
and its parts
0 
k
m
f 02
A 
(02   2 ) 2  4  2 2
2
1 
02  2
  tan 1 (
2 
)
2
0  
Designing a PAR
We can take what we have learned about the physical meanings
of the parts of the general solution to examine the ideal
conditions for a PAR system.
– General solution
– Damping coeff. and the quality factor
– The “notch”
Designing a PAR
A PAR is designed to exploit the harmonic term of the general
solution. To physically reason this, lets recall the period of a
sinusoidal function
T 
1
f
Where f is the frequency. When the frequency decreases, the
period of the wave increases.
The typical frequency range for bass notes is around 120-20Hz.
This means that bass notes tend to hang around longer
compared to higher frequencies.
This make them correspond to the harmonic term.
Designing a PAR
PAR’s work best with a low Q-factor. Increasing β is going to decrease
the Q-factor.
Q
0
2
Another way to think about desired Q and β, is with the transient
term. Oscillations die out in the absence of a driving force according
to the exponential in the term
x p  A' e  t cos(1   ' )
When designing a PAR you do not want the system to continue to
“ring” long after a note is over. We want this term to die out
quickly.
This means that a properly designed PAR should not have β << ω0 to
keep Q low.
Frequency vs. Output
• Red—ported cabinet
• Green—Sealed cabinet
• Blue—PAR (ω0 = about
15Hz)
20
• The PAR is being driven
above resonance here and
the notch occurs below the
range of human hearing
Further Exploration
• Effects of:
– box size on output
– Surface area of cone
– Resonant frequency on the notch (superposition)
• Why the “notch” occurs
Questions?
References
Passive Radiator Systems. (2010, August 16). Retrieved
Febuary 1, 2014, from DIY Subwoofers:
http://www.diysubwoofers.org/prd/
Transient (acoustics). (2014, January). Retrieved January
30, 2014, from Wikipedia:
http://en.wikipedia.org/wiki/Transient_%28acousti
cs%29
Taylor, J. R. (2005). Classical Mechanics. University
Science Books.
William E. Boyce, R. C. (2009). Elementary Differential
Equations and Boundary Value Problems. John
Wiley and Sons Inc.